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Harmonic trajectories are isochronous: all nonconstant oscillations with a fixed frequency have the same period, regardless of amplitude.

A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+ϵq^4$.

The quartic term is a specific anharmonicity of the oscillator potential.

Every finite nonabelian simple group has order at least 60; the alternating group A5 attains this bound.

The conjugacy classes of $A_5$ have sizesA normal subgroup is a union of these classes containing the identity. No proper nontrivial such union has size dividing $60$, so Lagrange's theorem proves that $A_5$ is simple.

The iteration $y_{n+1}=y_n-h\mathrm{sign}(y_n)$ agrees with the exact linear descent until the first step past zero. It then either remains at zero or alternates between two values of magnitude at most $h$. Consequently its uniform grid-point error is at most $h$ for arbitrarily long finite time intervals.

Lebesgue measure on the real numbers extends interval length: $\lambda ([a,b])=b-a$ for $a\le b$. Its restriction to the Borel sigma-algebra is the unique translation-invariant sigma-finite measure on that sigma-algebra normalized by $\lambda ((0,1])=1$. The full Lebesgue measure is the completion of a measure of this restriction; its domain is the Lebesgue measurable sets. Translation invariance follows by comparing a translated Borel measure on half-open intervals and applying the uniqueness theorem for measures. For the construction and interval properties, see MIT lecture 8.

Every singleton has Lebesgue measure zero: $\{x\}\subseteq [x-\epsilon ,x+\epsilon ]$ gives $0\le \lambda (\{x\})\le 2\epsilon$ for every $\epsilon >0$. Enumerating the countable set $\mathbb{Q}\cap [0,1]$ and applying countable additivity yieldsThus a dense set can be a null set, and a union of null sets indexed by an uncountable set can have positive Lebesgue measure. The distinction explains both the Dirichlet function example and why almost everywhere is weaker than everywhere.

A Dynkin system contains $\Omega$, is closed under complements, and is closed under countable disjoint unions. Equivalently, it contains differences $B\setminus A$ whenever $A\subseteq B$ are members.

A pi-system is a nonempty family of subsets closed under finite intersections.

A Borel set in a topological space belongs to the smallest sigma-algebra containing every open set.

A sigma-algebra atom is set-theoretically minimal. An atom is instead indivisible only modulo sets of measure zero.

An atom of a sigma-algebra is a nonempty measurable set containing no proper nonempty measurable subset. A finite sigma-algebra partitions its underlying set into atoms, and each measurable set is a union of them.

The Five lemma has a short form: in a commutative diagram of short exact sequences, if the maps on both outer terms are isomorphisms, then the map on the middle terms is an isomorphism.

A short exact sequence splits when its injection has a retraction, equivalently when its surjection has a section. In that case the middle module is isomorphic to the direct sum of the two outer modules.

Thermal relaxation drives the local sound speed toward an equilibrium value on a cooling time $\tau$. Fast relaxation gives an isothermal response, while slow relaxation gives an adiabatic response.

Sheaf cohomology consists of the right derived functors of global sections. The zeroth group is $H^0(X,\mathcal{F})=\Gamma (X,\mathcal{F})$, and higher groups measure obstructions to gluing local sections.

A sheaf of ${\mathcal{O}}_X$-modules is locally free of finite rank when every point has an open neighborhood $U$ on which it is isomorphic to ${\mathcal{O}}_U^{\oplus r}$ for some finite $r$. If $X$ is connected, the rank is constant.

A sheaf of modules is locally free of rank $r$ if each point has a neighborhood $U$ on which it is isomorphic to ${\mathcal{O}}_U^r$. Finite-rank locally free sheaves correspond to vector bundles in the standard geometric settings.

The inverse image sheaf $f^{-1}\mathcal{E}$ is the sheaf associated with the presheaf whose sections near $U\subseteq X$ are obtained as a colimit of $\mathcal{E}(V)$ over open sets $V\supseteq f(U)$.

This is the sheaf obtained by applying the inverse image functor.

For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image is defined by $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$ for every open $V\subseteq Y$.

The direct image sheaf is the result of the direct image functor: $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$.

The tensor product of two sheaves of ${\mathcal{O}}_X$-modules is the sheafification of $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$. Its stalk at $x$ is ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$.

A sheaf of modules on a scheme is quasi-coherent when every affine chart $\mathrm{Spec}A$ restricts it to the sheaf associated with an $A$-module. On an affine scheme it is determined by its module of global sections.